The sentence "他努力工作" omitted '地' in "他努力(地)工作" (he works in a hard working manner)
The sentence "他应该用功学习" omitted '地' in "他应该用功(地)学习" (he should study in a hard working manner)
Examples of 努力; 用功 as verb:
"他在努力" - "He is working hard"
"不要阻他用功" - "Don't stop him from working/ study hard"
Most common usage of 用功 is "to study hard"
Edit:
can you omit anytime with any adjective you use as an adverb?
It depend on the context
強烈 (strong)
強烈地 (strongly)
Look at the following two sentences:
"強烈反對" = [強烈(adj)+反對(n)] = "strong objection"
"強烈地反對" = [強烈地(adv)+反對(v)] = "strongly object"
If you omit 地 in sentence #2, it would become the same as sentence #1 . We would not know "強力反對" mean "strong objection" or "strongly object".
But with enough context, we can tell which is which.
"(他)強烈反對" = "(He) strongly objects" - With the subject "he", we know 強烈反對 must be [adverb +verb] . Therefore, we can omit the adverb marker '地' in "(他)強烈(地)反對"
"(他的)強烈反對" = "(his) strong objection" - With the possessive pronoun "his" , we know 強烈反對 must be [adj + noun]